This function returns the awards vector assigned by the reverse Talmud rule to a claims problem.
RTalmud(E, d, name = FALSE)The awards vector selected by the reverse Talmud rule.
If name = TRUE, the name of the function (RTalmud) as a character string.
The endowment.
The vector of claims.
A logical value.
Let \(N=\{1,\ldots,n\}\) be the set of claimants, \(E\ge 0\) the endowment to be divided and \(d\in \mathbb{R}_+^N\) the vector of claims such that \(D=\sum_{i \in N} d_i\ge E\).
The reverse Talmud rule (RTalmud) coincides with the constrained equal losses rule (CEL) applied to the problem \((E, d/2)\) if the endowment is less or equal than the half-sum of the claims, \(D/2\). Otherwise, the reverse Talmud rule assigns \(d/2\) and the remainder, \(E-D/2\), is awarded with the constrained equal awards rule with claims \(d/2\). Therefore, for each \(i\in N\),
$$\text{RTalmud}_i(E,d) = \begin{cases} \max\{\frac{d_i}{2}-\lambda,0\} & \text{if } E\leq \tfrac{1}{2}D\\[3pt] \frac{d_i}{2}+\min\{\frac{d_i}{2},\lambda\} & \text{if } E \geq \tfrac{1}{2}D \end{cases},$$
where \(\lambda \geq 0\) is chosen such that \(\underset{i\in N}{\sum} \text{RTalmud}_i(E,d)=E\).
Chun, Y., Schummer, J., and Thomson, W. (2001). Constrained egalitarianism: a new solution for claims problems. Seoul Journal of Economics 14, 269-297.
Thomson, W. (2019). How to divide when there isn't enough. From Aristotle, the Talmud, and Maimonides to the axiomatics of resource allocation. Cambridge University Press.
AA, allrules, APRO, Talmud, CEA, CEL, CD, RA.
E=10
d=c(2,4,7,8)
RTalmud(E,d)
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